Jul 1, 2018

36

#### Explanation:

When a line is a perpendicular bisector of another line, then that means that it divides the line into 2 equal parts. It also means that the lines are at right angles to each other
For example, if AC is a perpendicular bisector of DE, then DB=BE

If we know that DB=BE, AC is at right angles to DE and AB is a common side of $\triangle A B E$ and $\triangle A B D$, then we can assume that AD=AE.

Why? By using congruent triangles

In $\triangle A B E$ and $\triangle A B D$,
1. AB is the common side
2. DB=BE (AC is a bisector of DE)
3. AC is at right angles to DE (AC is a perpendicular bisector of DE)
then $\triangle A B E = \triangle A B D$ (SAS or two sides and one angle between the two sides are equal)

Therefore, AD=AE (same sides of proven congruent triangles are equal)

$3 x - 9 = x + 21$
$2 x = 30$
$x = 15$

Since we want to find side AE which is equal to $x + 21$, we can sub in $x = 15$ to find its value which is $15 + 21 = 36$

Jul 1, 2018

Since $\overline{A C}$ perpendicularly bisects $\overline{D E}$,

• the lengths of $\overline{D B}$ and $\overline{B E}$ are the same (since $\overline{D E}$ is divided in two equal parts), and angle $\angle A B E$ is ${90}^{\circ}$ (since $\overline{A C}$ intersects $\overline{D E}$ perpendicularly).
• triangle $\Delta D B A$ is a horizontal reflection of triangle $\Delta A B E$ (since $\overline{A C}$ divides one triangle $\Delta D A E$ into two identical halves).

It then follows from the triangles being a reflection of each other that the lengths of $\overline{A D}$ and $\overline{A E}$ are identical. Therefore:

$3 x - 9 = x + 21$

$\implies 2 x - 9 = 21$

$\implies 2 x = 30$

$\implies x = 15$

As a result, the length of $\overline{A E}$ is:

$\textcolor{b l u e}{L} = x + 21 = 15 + 21 = \textcolor{b l u e}{36}$